Application of the boundary-value technique to singular perturbation problems at high Reynolds numbers

JOLKNAL 0,. MATHEMAI‘I(‘AI. ANALYSIS AND APPLIC’ArlOYS 122. 70 87 (1987)

Application

of the Boundary-Value

Technique

to Singular

Perturbation

Problems

at

High Reynolds

Numbers

R. Y. QASSIM

Escola de Engenharia and CoordenaQo cl0.v Progrumus de Ptis -grudua~Zo em Dzgcwhtrrirr, Universidadc Federal do RIO dc Janeiro, C. P. 6XSO3.

Rio dc~ Juneiro, Brazil

AND

A. P. SILVA FREIRE

Coordenuqfo dos Programas de Pds-graduuqcio em Engenharia, Universidade Federal do Rio de Janeiro, Brazil;

pre.wntlJ: ai the Department of Engineering, Universiiy of Cambridge, Cambridge, CB3 9Ell, England

Submitted by C.-C. Rota Received July 31, 1985

The boundary-value technique, advanced by Roberts for the solution of singular pertubation problems of ordinary differential equations where the small parameter multiples the highest derivative, is extended to the solution of the Navier-Stokes equation at high Reynolds numbers. Three standard flows-uniform flow past a plate, flow with a linearly adverse external velocity, and shear flow past a flat plate-have been chosen as test problems with a view to evaluating some of the features of the boundary-value technique, particularly in comparison with coef- ficient matching techniques as examplilied by the method of matcher asymptotic expansions. (” 19X7 Academic Press, Inc

1. INTRODUCTION

Singular perturbation problems pervade fluid mechanics in a variety of forms and contexts. In flow Reynolds number flow the singularity arises due to the fact that inertial effects are small but not negligible far away from the body, and in boundary layer flows at high Reynolds numbers, the perturbation is singular due to the fact that viscous effects can be neglected everywhere except within a thin layer next to the body. A number of techniques have been developed and applied to singular perturbation problems in general and in fluid mechanics in particular, see Cl, 2, 33. The

70 0022-247X/87 $3.00

C‘opyrght c 1987 by Academic Press, Inc. All rights of reproduclmn in any form reserved.

THE BOUNDARY-VALUE METHOD IN FLUID FLOW 71

common feature of those techniques is that the flow is divided into two regions, called inner and outer, in each of which an asymptotic expansion

is determined, and these two expansions are combined to obtain a

uniformly valid solution, the precess of combining the pair of expansions

being performed through matching of their corresponding coefficients. Two

aspects stand out in those so-called coefficient matching techniques

(CMT): selecting the appropriate asymptotic expansions for the specific

problems under consideration and matching the expansions once they have

been determined. The choice of the appropriate asympotic expansions is far

from straightforward as it involves a good measure of experience, insight,

and intuition. Further, the matching of the expansions through their coef-

ficients involves a numbers of conceptual difftculties.

With a view to avoiding these difficulties altogether, Roberts [4] has

presented a method, which he denotes the boundary value technique

(BVT), for the solution of singular perturbation problems of ordinary dif-

ferential equations, where the small perturbation parameter multiplies the

highest derivative. In the boundary-value technique, the interval over

which the problem is formulated is divided into two intervals, inner and

outer, in each of which a solution is obtained by analytical, numerical, or asymptotic methods. The outer solution, corresponding to the outer inter- val, constitutes the terminal condition for the inner solution which is valid in the inner region. The inner problem is then solved iteratively for various

values of the terminal point of the inner interval until the successive

iterations of the solutions are close to each other within the desired degree

of accuracy. Herein lies the principal difference between the coefficient

matching techniques on the one hand the boundary-value technique on the

other. The adjoining of the outer and the inner solutions in the BVT is

carried out at a point in the domain of the problem, this point being found

iteratively, while in the CMT the inner and the outer expansions are

matched asymptotically.

In addition to the above-mentioned advantages of avoiding altogether

the far from simple problem of the choice of asymptotic expansion and the

intricate question of matching the inner and the outer expansions, in

general there are a number of significant computational differences

associated with the BVT in relation to the CMT. In the first place, once the original problem is divided into its inner and outer parts, one is faced with the task of solving this pair of problems in the BVT and a sequence of

problem pairs (zero-order, first-order, etc.) in the CMT. Generally, less

computational effort would be involved in the former than in the latter

situation, although it should not be forgotten that the BVT involves an iterative scheme and that the problem pair arising in the BVT is normally more complex than the problem pairs corresponding to the various orders

compared with the CMT arises when dealing with problems involving infinite intervals. In the BVT, one has to solve the outer problem as an initial-value problem and the inner problem as a multi-point boundary- value problem over a,finite interval. This is in contrast to the situation that arises in the CMT, where the outer problems are initial-value but the inner problems are multi-point boundary-value over an infinite interval. Clearly, it is simpler mathematically and computationally to deal with multi-point boundary-value problems in finite rather than in infinite domains.

The objective of this paper is to extend the application of the BVT to some singular pertubations of the Navier-Stokes equations. In order to evaluate the results, we have selected flows for which solutions have been obtained by classical boundary layer theory and CMT. In Section 2, we give a general description of the BVT as applied to singular perturbation problems of partial differential equations. This is done since Roberts [4] presented the technique in the context of ordinary differential equations. Two boundary layer-type flows are determined in Section 3, using the BVT, and the results are compared with the established solutions of Blassius and Howarth [S] for uniform flow past a flat plate and flow with a linearly adverse external velocity, respectively. In Section 4, a hybrid method, involving the BVT and the technique of asymptotic expansions, is utilized to determine shear flow past a flat plate. This problem is also solved by use of the BVT alone and results are compared with the solutions of Murray [6]. This is done with a view to demonstrating the potential of the BVT in so far as its couplability with other methods and suggestions are presented with respect to the extension of the BVT to more complex flows and to deal with other types of singularities in fluid mechanics.

2. DESCRIPTION OF THE BOUNDARY-VALUE METHOD FOR

SINGULAR PERTURBATION PROBLEMS

Consider a partial differential equation of the form

,f(E, -x, I’>-, PL, pr> pL, ,“., Px,, &,v,...) = 0, (1) where the small parameter E multiplies a term containing one of the highest derivatives, in x say. We seek a solution to Eq. (1) satisfying m boundary conditions in x, the validity of this solution being for small values of the parameter E.

Letting F --f 0 results in a lowering of the order of the equation, and we obtain the so-called outer equation

Jot? YY> K PL,, PL,,.-, Pry, P,,,,...) = lim f(s,

c-0’

THE BOUNDARY-VALUE METHOD IN FLUID FLOW 73

Clearly, Eq. (2) can only be expected to satisfy m - n boundary con- ditions in x. Solving Eq. (2) subject to those m-n boundary conditions con- stitutes the outer problem.

Returning now to Eq. (l), we define a new independent variable o by the transformation

X

7=go’ (3)

where g is an arbitrary scale function to be selected. Using Eq. (3) we may rewrite Eq. (1) as

.fitE, 7, Y,..., CL, PT, PLv,..., PL,T, PT.* ,...) = O (4)

this being the so-called inner equation. Equation (4) is then to be solved subject to m boundary conditions in x, n of these being given by

PObf, Y,...) = $bf, yv.. 17

pO(x,, Y,...) = &f, Y,...),

where

xi

v=g(E)1

(5)

(6)

and x, is an arbitrary value of x. Clearly, r/-is a function of the independent variables y,... . Solving Eq. (4) subject to the m-n boundary conditions and to Eq. (5) constitutes the inner problem.

The overall solution procedure then consists of the following steps:

(i) Solution of the outer problem.

(ii) Solution of the inner problem for arbitrary 7,.

(iii) Solution of the inner problem for slightly different rf

(iv) Verification of the proximity of successive iteration for the

solution of the problem.

Several comments are now in order. The crux of the boundary value

technique lies in the determination of the terminal boundary condition for

the inner solution. This is achieved by an iterative process, in which an

arbitrary point 7,. is chosen as the point where the inner solution ends and

the outer solution begins. In more physical terms, tf may be interpreted as

the thickness of the boundary layer region arising in the singular per- tubation problem.

74 CJASSIM AND SII.\‘A IXEIKfi

3. APPLICATION OF THE BOUNDARY-VALUE METHOD

TO VISCOUS FLIJD FLOW

In order to evaluate the applicability of the boundary-value method to

the NavierStokes equations, we have chosen two problems involving flows

of a boundary layer nature and whose solutions via classical boundary layer theory have been obtained with some precision. These are uniform flow past a semi-infinite flat plate and flow with a linearly adverse external velocity. We consider these problems in turn.

(a) UNIFORM FLOW PAST A SEMI-INFINITE FLAT PLATE. Consider steady

uniform two-dimensional incompressible viscous fluid flow past a semi-

infinite plate. Using rectangular Cartesian coordinates with origin at the

leading edge of the plate, the x-and y--coordinate axes parallel and per-

pendicular to the plate, respectively, the vorticity transfer equation may be written in terms of the stream function Y as

YL.Y’,,,

+ y, yn- y/,y,.,,.- y’,yw

=e(YU,,,,.+2Y,.,,.+

y ,‘,‘,,.

).

where E is the inverse of the Reynolds numbers; i.e., E = 1/R = v/UL. We

seek a solution to Eq. (7) for small s’, subject to the no-penetration and

no-slip boundary conditions at the plate; i.e.,

I’ = 0, x>o; Y,= Y, =o, (8)

and to the uniform parallel flow boundary conditions faraway from the

plate; i.e.,

I’+“, x>o; Yy=O, YL.=l. ₍₉₎

The outer flow equation, obtained by setting E = 0, in Eq. (7), is given by

y.,. yy,,.,. + yvl ‘YYY, - y’, yvv, - Y.Y yxrj, = 0, (10)

whose solution is required to satisfy the no-penetration boundary condition

at the plate; i.e.,

y = 0, x>o, Y’,=O, (11)

or equivalently and without any loss of generality

Y = 0, x>o, Y=O. ₍₁₂₎

THE BOUNDARY-VALUE METHOD IN FLUID FLOW 75

ditions at infinity, given by Eq. (9). We note that as the outer Eq. (10) is of

third-order and the equation of motion (7) is of fourth-order, the former

can satisfy only three of the four boundary conditions given by Eqs. (8) and (9). Based on physical considerations, the no-slip boundary condition at the plate, given by Eq. (8), is abandoned in the outer flow problem. The solution of Eq. (10) subject to boundary conditions (9) and (12) is uniform flow; i.e.,

!P=y. ₍₁₃₎

In order to obtain the inner flow equation, we adopt the following

variable transformations

Y

5=d(E)) (14)

vx, Y) = d(E) x(x, z), (15)

where d(s) is a yet to be determined function of E. The function d(e) is determined in the same manner as in the CMT; i.e.,

A(&)=& 112 _ - R-l/2 , (16)

and then Eq. (7) may be written as

xrx.x,,

-X\-Xr~,+&CX,X,,,--X,X,,,l

= XT,,, +

z =o, x > 0; x.y = x* = 0, (18)

5 = Tf, x>o; x,=0, XT= 1, (19)

which are obtained by transforming the boundary conditions (8) and (9)

respectively.

As a first approximation, in the limit as E tends to zero, or equivalently

when R tends to infinity, Eq. (17) becomes.

XrXrrr = XxXrrr = Xrrrr, subject to boundary conditions (18) and (19).

Using the similarity transformation

x(x, r) = x”‘f(rlh

r/(x,z) = x”27,

(20)

(21) (22)

Eq. (20) becomes

f’.f’” +.l”” + 2.f”’ = 0,

and the boundary conditions ( 18) and (19) become respectively,

(23)

‘1 =o,

,f = f’ = 0,

‘1 =v/,

.f=rlpf’=

1.

We note that Eq. (23) is the derivative of the Blassius equation of classical

boundary layer theory. Furthermore, three of the boundary conditions (28)

and (29) are derivatives of the corresponding boundary conditions of the

Blassius problems, the exception being (25), which cannot be prefixed in

the Blassius problem as it is of third order. In fact, from the Blassius

solution, we have

where [S],

b = 1.7208.

This difference is due to the fact in the BVT as applied to this problem, the displacement effect is not taken into account as opposed to the solution obtained by matched asymptotic expansions (MAE). The two-point boun-

dary-value problem given by (23)-(25) has been solved by orthogonal

collocation for increasing values of qf, and convergence is obtained at

yll= 250. The function

f

solution of the Blassius problem [S]. It is seen that agreement is quite

satisfactory, with a maximum percentage difference of 8 % between the two

solutions.

(b) FLOW WITH A LINEARLY ADVERSE EXTERNAL VELOCITY, Consider the corresponding case to (a) when the fluid velocity profile faraway from the flat plate is given by

U= U,= Ax, ₍₂₆₎

where U, and A are constants. Following the procedure shown in case (a),

the equation of motion is given by (7), subject to the boundary conditions (8) and

Y-+~, x>o; y/,=0, !P,,= 1 --ax, (27)

TABLE I Uniform Flow Past a Flat Plate: U(x)= Li, ‘If 10 100 250 Blassius F2 2 4 f(v) f’(v) f(v) f'(rl) f(v) f’(s) f (exact) fi.,.m, %f %f’ 0.0 0.0 0.0 0.0 g 0.0 0.0 0.0 0.0 0.0 0.0 0.0 < 1.0 0.315183 0.608609 0.182794 0.362053 0.177615 0.353367 0.16557 0.32979 7.27 7.15 ’ 2.0 1.154978 1.030385 0.710180 0.682052 0.694724 0.669526 0.65003 0.62977 6.88 6.31 s 3.0 2.289017 1.198203 1.518564 0.917530 1.483860 0.88767 1.39682 0.84605 6.23 5.05 5 4.0 3.495045 1.197923 2.507571 1.041282 2.433052 0.992896 2.30576 0.95552 5.52 3.91 g 5.0 4.670909 1.152519 3.567016 1.064762 3.444793 1.022555 3.28329 0.99155 4.92 3.13 Kj 6.0 5.801540 1.110177 4.626192 1.054712 4.469697 1.0259 16 4.27964 0.99898 4.44 2.70 7.0 6.893947 1.075769 5.678636 1.05053 1 5.495788 g 1.025845 5.27926 0.99992 4.10 2.59 8.0 7.954936 1.046973 6.727265 1.046619 6.521067 1.024752 6.27923 1.OOOOO 3.85 2.47 2 9.0 8.989178 1.022057 7.771878 1.042788 7.545400 1.023918 7.27029 100000 3.66 2.39 3 10.0 9.989178 1.OOOOOO 8.813518 I .040891 8.568825 1.02292 1 8.27929 l.OQOOO 3.50 2.29 2 11.0 10.989178 1 .oooooO 9.854043 1.039942 9.591351 1.022162 9.27929 1 .ooooo 3.36 2.21 w 12.0 11.989178 l.ooooOO 10.892929 1.037838 10.613153 1.021437 10.27929 1 .ooooo 3.25 2.14 ? 13.0 12.989178 l.OOcQOO 11.930001 I .036480 11.634237 1.020754 11.27929 l.OOOOO 3.15 2.07 14.0 13.989178 1.000000 12.966112 1.035759 12.654718 1.020226 12.27929 1.0#00 3.06 2.02 2

7x QASSIM AND SII.VA FREIRk

which is required to satisfy boundary condition (12) and the far flow boun-

dary condition faraway from the plate, given by (27). The outer flow

solution is given by

Y= (1 -ax) J’. (28)

The inner flow problem is formulated in a manner analogous to case (a), and the resulting problem in the limit as E tends to zero can then be written as

XrLrr - X.rXrrr = xirrr,

t = 0,

x.r = XT = 0,

T = T,,

Utilizing the similarity transformation

X(-~,t)=x”“{f,(r?)-(8x)f,(ul)+(8x*).f*(r)-

+ ... 1,

~ l/2

q(x, g=y, (31)

x*(x)=$

0

in Eq. (22), and separating the terms of the same order in x, we obtain for the two lowest order equations, respectively

.f&fo"+fofd"+f'"=O,

-fofi”+f{fb” + 3fd”

+ j; +fof? +f;"=o.

The corresponding boundary conditions are given by

rl =o,

.fo=fb =o,

9 = v/2

fo = %fb=Z

rl =Q

fl =f; =o,

It is worth noting that (33) and (34) are derivatives of the Howarth lowest order pair of equations, respectively. These, being of third-order,

TABLE II Linearly Adverse External Velocity Flow: U(x) = U, - Ax 4 20 40 Howarth rl f,(v) f;(v) fibl) f;(v) f,(v) f;(q) f,(o) f;(v) %fi %f; 0.0 O.OOOOOO 0.000000 O.OOOOOO O.OCQOOO O.OOOOOO OOOOOOO O.OOOOO O.OOOOO 0 0 0.2 0.006901 0.063568 0.014935 0.143873 0.015436 0.148703 0.01908 0.18411 19.1 19.2 0.4 0.024850 0.118471 0.055294 0.254026 0.057150 0.262554 0.07098 0.32823 19.5 20.0 0.6 0.053315 0.164761 0.114308 0.330710 0.118146 0.341811 0.14771 0.43250 20.0 21.0 0.8 0.090196 0.202706 0.185535 0.376820 0.191763 0.389465 0.24137 0.49768 20.6 21.7 1.0 0.133894 0.233129 0.263317 0.397179 0.272154 0.410502 0.34432 0.52596 21.0 22.0 1.2 0.183061 0.257792 0.343092 0.397827 0.354605 0.411165 0.44959 0.53195 21.1 21.2 1.4 0.236831 0.279781 0.421570 0.385304 0.435713 0.398213 0.55143 0.49309 21.0 19.2 1.6 0.295029 0.302539 0.496759 0.365932 0.5 13420 0.378180 0.64581 0.44900 20.5 15.8 1.8 0.357896 0.325502 0.567849 0.345104 0.586889 0.356640 0.73069 0.39972 19.7 10.8 2.0 0.424593 0.339269 0.634958 0.326565 0.656239 0.337464 0.80592 0.35367 18.6 4.6 2.2 0.492826 0.341416 0.698727 0.311699 0.722134 0.322083 0.87273 0.31608 17.3 1.9 2.4 0.560549 0.334355 0.759778 0.298814 0.785217 0.308747 0.93305 0.28879 15.8 6.9 2.6 0.635989 0.318626 0.818039 0.282820 0.845412 0.292197 0.98889 0.27098 14.5 7.8 2.8 0.687626 0.297129 0.872674 0.263482 0.901856 0.272190 0.04192 0.26045 13.4 4.5 3.0 0.744790 0.297129 0.923575 0.246206 0.954436 0.2543 11 1.09338 0.25480 12.7 0.2 3.2 0.798035 0.274887 0.971509 0.234154 1.003945 0.241829 1.14403 0.25203 12.2 4.0 3.4 0.849243 0.259436 1.017658 0.228374 1.051603 0.235825 1.19430 0.25079 11.9 6.0 3.6 0.899744 0.253744 1.063224 0.228305 1.098671 0.235721 1.24440 0.25029 11.7 5.8 3.8 0.949946 0.251552 1.109251 0.232287 1.146172 0.239802 1.29443 0.25010 11.5 4.1 4.0 1.OWOOO 0.250577 1.156277 0.238076 1.194716 0.245746 1.34444 0.25003 11.1 1.7 4.2 1.049950 0.250000 1.204446 0.243345 0.244433 0.251151 1.39444 0.25001 10.8 0.5 4.4 1.099764 0.248471 1.253452 0.246200 1.295008 0.254060 1.44445 0.25000 10.3 1.6

80 QASSIM ANI) SILVA I-KEIKE

(38) is not prefixed in the Howarth solution, but rather is given from the solution as

‘I = II,:

.f, =; +

This difference, as in the Blassius problem, is due to the fact that in the BVT solution, the displacement effect is neglected, while this is not the case in the MAE solution. The problem, given by (33) (35), and (36) and the

problem given by (34), (37), and (38) have been solved by orthogonal

collocation for decreasing values of qr, convergence being obtained at

g, = 250 and ql-=40, respectively. The function f,, is shown in Table II, along with the solutions of Howarth.

4. APPLICATION OF A HYBRID BOUNDARY-VALUE-COEFFICIENT

MATCHING METHOD TO VISCOUS FLUID FLOW

In order to demonstrate the flexibility and the versatility of the BVT in

the solution of fluid flow problems, we consider the classical problem of shear flow past a flat plate. Solutions have been obtained for this problem through the use of matched asymptotic expansions by Murray [6], and we

solve this problem in two ways: first by application of the BVT on its own,

and then by the joint application of the BVT along with matched

asymptotic expansions.

Consider steady two-dimensional incompressible viscous lluid flow past a

flat plate, where the fluid is in linear shear faraway from the plate. In rec- tangular Cartesian coordinates, the vorticity transfer equation is (7), which we wish to solve for small E, subject to the boundary conditions (8) at the plate, and to linear shear flow faraway; i.e.,

y-‘=J, x>o; YJ,=o, !ty,= 1 +u’o,r. (40)

As seen in Section 3, the outer flow equation is given by (lo), which we now wish to solve subject to the boundary conditions at the plate given by

(12), while the boundary condition (40) has to be satisfied faraway from

the plate. The outer flow solution can then be immediately written as

I=y+*.

THE BOUNDARY-VALUE METHOD IN FLUID FLOW 81

we wish to solve subject to the boundary conditions (18) at the plate, and faraway from the plate, we have

t = Tf, x>o; x,=0, XT= 1 +woy. ₍₄₂₎

Using the expansion

X=X0+&X’, (43)

in Eq. (17) and in boundary conditions (18) and (42), we obtain for x0

x:x”,,, - x”, x:7, = XL ) (44)

z = 0, X:=X”,=% (45)

z = t/, x:= 1, x”,=o, ₍₄₆₎

and for x ‘, we have

x:x.:,, + i&J’ -xOxL - xR,xt, = XL (47)

z = 0, x; = xt, = 0, (48)

2=5 1’ xf=wot, x:=0. ₍₄₉₎

Applying the transformations

x0 = x’%rl),

x’ = 4woxdrlh (51)

qq, ₍₅₂₎

the problems given by (44)-(45) and (47)-(49) become respectively

f”’ + f’f” + ff”’ = 0, ₍₅₃₎ ?j=o, f=f’=O, ₍₅₄₎

v=vf,f=2rl,f'=2,

g'"+g'f" +fg"'+

x2 QASSIM AND SILVA l-REIRb

solution is shown in Table I, and the problem given by (55)-(57) has been solved by orthogonal collocation and the results are displayed in Table III.

The above solution results from the direct application of the BVT to the problem of linear shear flow past a flat plate. With a view to evaluating the versatility of the BVT, we utilize this method jointly with that of matched asymptotic expansions (MAE) to solve this problem in an alternative manner. For this purpose, we introduce the asymptotic expansions for the inner and the outer stream functions.

Y(x,y) = (x,y) + &‘ciqx,y) + “. ) ₍₅₉₎

X(x,T)=E”‘X”(x,T)+EX’(X,T)+ .” ₍₆₀₁

Substituting Eq. (59) in the outer flow Eq. (IO), and equating the coef- ficients of the various powers in E to zero, we obtain for the zero-order outer equation

VP= wo, ₍₆₁₎

where

)$, = )p + E’P1.l?’ + . . )

(62)

subject to the boundary conditions

4’ =o,

x> 0, P=o,

p+~, x>o, Y$+1+w,y. (64)

The solution of Eq. (61) subject to boundary conditions (63) and (64) is given by

p=,+$.

Substitution of Eq. (65) in the inner flow Eq. (17) and equating the coeffkients of the resulting power expansion in to zero, we have for the first-order inner equation

Since Eq. (66) constitutes an exact differential, we have

TABLE III Shear Flow Past a Flat Plate: U(x) = U,, + won 4 - rl g(rl) g’(q) g(4) g’(q) g(v) g’(v) Rexact O.OOOOOO O.OOOOOO O.OOOOOO 0.000000 0.000000 O.OOOOOO O.OOOOOO 0 0 0.2OOOoO 0.01553 1 0.159806 0.023287 0.234236 0.014750 0.154011 0.06021 0.59050 0.4OOoOO 0.065669 0.345729 0.094107 0.474760 0.064156 0.346205 0.23139 1.10870 0.6OOOOO 0.155354 0.554576 0.213284 0.717130 0.155552 0.573142 0.49837 1.54698 O.XOOOOO 0.288707 0.781502 0.380827 0.957639 0.295370 0.829390 0.84423 1.89611 1.oooooO 0.468785 1.020875 0.595956 1.192403 0.488828 1.108420 1.25044 2.14999 1.2OOOoO 0.697525 1.267373 0.857182 1.418107 0.739820 1.403886 1.69800 2.31087 1.4OOoOO 0.975999 1.517870 1.162414 1.632076 1.051253 1.712142 2.16951 2.39299 1.6OOOOO 1.304779 1.769836 1.509106 1.832473 1.424998 2.024987 2.65164 2.42199 l.XOOOOO 1.683638 2.017113 1 .X94442 2.018482 1.860557 2.327314 3.13689 2.42963 2.OOOOOO 2.110397 2.245583 2.315563 2.190503 2.353921 2.599537 3.62408 2.44577 2.2OooOO 2.578981 2.431047 2.769827 2.350338 2.896923 2.820021 4.11712 2.49144 2.4OOOOO 3.078963 2.559695 3.255108 2.501384 3.477109 2.969040 4.6236 2.57583 2.600000 3.599903 2.644530 3.770135 2.648819 4.079715 3.046559 5.15005 2.69786 2.XOOOOO 4.135753 2.714089 4.314867 2.799798 4.692622 3.077972 5.70444 2.85023 3.OOOOOO 4.686547 2.798345 4.890906 2.963632 5.310626 3.105475 6.29155 3.02377 3.2OOOOO 5.257936 2.927100 5.501683 3.147017 5.937873 3.178835 6.91480 3.21037 3.4OOcQO 5.862902 3.136255 6.150568 3.343321 6.587289 3.324842 7.57617 3.40417 3.600000 6.516895 3.411315 6.839391 3.544397 7.270156 3.507016 8.27670 3.60154 3.XOOOOO 7.228926 3.709266 7.568788 3.748453 7.990799 3.700052 9.01689 3.80052 4.OOOOOO X.OQOOOO 4oOOOO0 8.338647 3.949690 8.750414 3.896881 9.79695 4.00015 4.2OOOOO 8.829846 4.307734 9.148462 4.147936 9.549794 4.097511 10.61697 4.2OQO4 4.400000 9.73 1920 4.752436 9.997626 4.343284 10.389481 4.299171 11.47697 4.40000 13 - 17 - Murray % g % g’ 0 0 75.50 73.92 72.27 6X.77 6X.79 62.95 65.01 56.26 60.91 40.45 56.43 39.25 51.54 2X.45 46.26 16.39 40.69 4.21 35.05 6.29 26.64 13.19 24.79 15.27 20.78 12.93 17.74 7.99 15.59 2.70 14.13 0.98 13.05 2.33 12.16 2.62 11.38 2.64 10.68 2.58 10.05 2.44 9.48 2.29

84 QASSIM AND SILVA FREIRE

where p,(x) is an arbitrary function of .Y. Using the Matching Principle, we have

x:(x> w)= 1, LJ,(x)=O.

From boundary condition (8), we obtain

x:(x, 0) =x:(x, 0) = 0.

Utilizing the transformation

x’(x, T) = x”2f(ij), xli2 q=Tp, (68) (69) (70) (71) (72) we obtain from Eq. (67)

ff” +f”’ = 0, ₍₇₃₎

and from boundary conditions (68) and (70), respectively

tl=o,

f=f'=O,

v = vl/,

f'=2

Returning to the outer flow Eq. (lo), the first-order external equation

may be written as

v2!P =o, (76)

subject to the boundary conditions

y = 0,

x<o; Y’=O,

y

y-+a, x>o, Y”=o. (79)

The boundary condition (78) is obtained by the Matching Principle. The

solution of (76) subject to (77)(79) is given by

!P’ = -p Re dx. ₍₈₀₎

The first-order inner equation may be written as

THE BOUNDARY-VALUE METHOD IN FLUID FLOW 85 Since Eq. (8 1) is an exact differential, we have

x9xX

+ xx, - xx, - xx = xi,, +P*(x),

where p2(x) is an arbitrary function of x. Matching yields

x; (4 cc) = ‘SWIJ,

p2(x)=wox;(x, ++wx.

From the boundary condition (8) we have x:(x, 0) = x.f. (x, 0) = 0.

(82)

(83) (84)

(85)

Using the transformation

x0(x, 7) = x1’2f(s),

x1(4 7) = 4woxg(rl),

ul=

and from boundary conditions (83) and (85) we have

‘? = 0, g=g’=O, (90)

? = VJ/, g’ = 11. (91)

The problem given by Eq. (89) subject to boundary conditions (90) and

(91) is identical to the one fomulated and solved by Murray [6]. These

results are also shown in Table III for the purpose of comparison. It is

readily observed from Table III that the differences between the solution

obtained directly from the BVT and that based on the BVT-MAE hybrid

method or equivalently that of Murray, are significant, particularly if com-

pared with our corresponding situation in the two former flow problems. Apart from the fact that the ordinary differential equations are different for

the different problems under consideration and the consequent possible

solution behaviour, we believe there are some important differences in the

application of the BVT and the BVT-MAE method to this problem. Essen-

tially, in the BVT, we have account for the external vorticity only, while in

the BVT-MAE account has been taken of both the external vorticity and

86 QASSIM AND SILVA I’RHRE

5. CONCLUDING REMARKS

The boundary-value technique, formulated and applied to singular per-

turbation problems of ordinary differential equations, has been extended to

the solution of the Navier-Stokes equations for three standard high

Reynolds number flows. These flows have been selected as solutions for they have been already obtained with confidence by classical boundary

layer theory and the technique of matched asymptotic expansions, and

hence can serve as an adequate test for the accuracy of the boundary value technique. In the following, we summarize our findings.

(i) The simplicity of the BVT, demonstrated by Roberts for

ordinary differential equations, is shown to be valid for partial differential equations as well. Thus, the question of which asymptotic expansion to use which is inherent in all CMT’s, simply does not arise in the BVT. However,

in the transformation of variables, the scale function has still to be chosen a

priori based on additional physico-mathematical arguments. This point is

not touched upon by Roberts.

(ii) The flexibility of the BVT in that it can be used in a hybrid

fashion with another technique, shown by Roberts for ordinary differential

equations, is demonstrated in the problem of shear flow past a flat plate.

On the other hand, when coupled to other techniques, the advantage of

simplicity inherent in the BVT may be lost. This is demonstrated when the

BVT is used in combination with the method of matched asymptotic

expansions in the solution of the shear flow problem. Roberts does not

appear to have recognized this limitation when applying the BVT in

conjunction with other methods.

(iii) The fact that in the BVT the inner and the outer solutions are

matched at a specific and definite location means that the question of the matching principle just does not arise. This advantage is shown in the three flows which have been determined by use if the BVT alone. Furthermore, the specific location at which the matching is performed provides a direct and precise measure of the boundary layer thickness.

(iv) From the analytical and the computational points of view,

there is a substantial advantage in utilizing the BVT in that the inner flow

is determined from the solution of a two-point boundary-value problem

over a finite domain. This is in contrast to the CMT, for example the method of matched asymptotic expansions, where the inner flow is posed

as a two-point boundary-value problem over an infinite domain. This

feature is clearly demonstrated in the three flows considered, and this

THE BOUNDARY-VALUE METHOD IN FLUID FLOW 87

(v) The mathematical question of convergence of the BVT remains

open as noted by Roberts. Barring the settling of this question, we believe

it necessary to extend the present investigation to more complex flows than

those studied in the present work, albeit maintaining the approach of

selecting flows for which reliable solutions have been obtained by CMT’s and other established methods. In the way of examples, we mention low Eckman number, rotating flows, corner boundary layers, and trailing edge flows.

(vi) The criterion which Roberts utilized to terminate the iteration,

which has also been used in the present work, depends on the matching of the inner and outer flow variables only and nothing is said about the derivatives. Clearly, better matching should be obtained if a more complete criterion is envisaged, involving the first few derivatives, up to the order of the equation say, as well as the flow variables themselves.

(vii) When applied to partial differential equations, the joining of

the inner and the outer solutions is performed along a curve or surface as

the case may be, due to the increase in dimensionality of the problem as

compared to the case of ordinary differential equations.

(viii) As advanced by Roberts, the BVT is applicable only to singular

perturbation problems of the type where the small parameter multiplies the

highest derivative. Then there is clearly room for extending the method to

other types of singularities, particularly if we wish to apply the method to

low Reynolds numbers flows amongst others.

(ix) As Roberts has stated, the question of which boundary con-

dition to drop in the outer flow problem still has to be based on physical arguments as is the case in coefficient matching methods.

REFERENCES

I. J. KEVORKIAN AND J. D. COLE, “Perturbation Methods in Applied Mathematics,” Springer- Verlag, New York/Berlin, 1981.

2. A. NAYFEH, “Perturbation Methods,” Wiley, New York, 1973.

3. M. VAN DYPE, “Perturbation Methods in Fluid Mechanics,” Academic Press, New York, 1973.

4. S. M. ROBERTS, A boundary-value technique for singular perturbation problems, J. Math. Anal. Appl. 87 (1982) 489-508.

5. L. HOWARTH, On the Solution of the boundary layer equations, Proc. Roy. Sot. Edinburgh Sect. A. 164 (1983) 547-579.

6. J. D. MURRAY, “The boundary layer on a flat plate in a stream with uniform shear”, .I. Fluid Mech. 11 (1961) 3099316.